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Binary and Hexadecimal Numbers

CSE 1310 – Introduction to Computers and ProgrammingVassilis Athitsos

University of Texas at Arlington

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Decimal Numbers• A decimal number is an integer written the way we are used

to write numbers, using digits from 0 to 9.• Why do we use 10 digits, and not 2, or 20, or 60, or 150?• It is a result of:

– Having 10 fingers in our hands, which makes it easier to count up to 10.

– Cultural convention.

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Babylonian Numbers

• Babylonians used 60 digits instead of 10.– Make sure you know them for the exam :)

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Babylonian Numbers

• Babylonians used 60 digits instead of 10.– Make sure you know them for the exam :)– In case you are reading this at home: the line above is a

joke!

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Babylonian Numbers

• Babylonians used 60 digits instead of 10.• While we do not use that system for writing numbers

today, it has survived in our modern culture. Where?• In the way we measure time.

– 24 hours.– 60 minutes.– 60 seconds.

• Also in the way we measure angles.– 360 degrees.– 60 minutes.– 60 seconds.

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Binary Numbers• Computers, in their electronic circuits, essentially use two

digits: 1 and 0.• 1 corresponds to a certain electrical (or magnetic, or optical)

signal.• 0 corresponds to another electrical (or magnetic, or optical)

signal.• This means that we need to represent all numbers using those

two digits.– Numbers represented using 1 and 0 are called binary numbers.

• Java (like any modern programming language) allows us to use decimal numbers.

• Inside the computer, before the program gets executed, all decimal numbers are translated to binary.

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Binary Numbers

• Since 1310 is our introductory course, it is a good place for you to learn some basics about binary numbers.

• We will do a simplified version.• We will not worry about representing:

– Floating point numbers.– Negative numbers.

• We will learn how to represent positive integers in binary.

• We will learn how to translate from binary to decimal and the other way around.

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Reading a Binary Number

• In general, a binary number is represented like this:

bn-1 bn-2 … b2 b1 b0

• Each digit bi is either 1 or 0.• For example, consider binary number 10011.• What is n?• What is b0?

• What is b1?

• What is b2?

• What is b3?

• What is b4?

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Reading a Binary Number

• In general, a binary number is represented like this:

bn-1 bn-2 … b2 b1 b0

• Each digit bi is either 1 or 0.• For example, consider binary number 10011.• What is n? 5• What is b0? 1

• What is b1? 1

• What is b2? 0

• What is b3? 0

• What is b4? 1

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Reading a Binary Number

• In general, a binary number is represented like this:

bn-1 bn-2 … b2 b1 b0

• This binary number can be translated to decimal using this formula:

bn-1*2n-1 + bn-2*2n-2 + … + b2 * 22 + b1 * 21 + b0 * 20.

• As a quick review: – What is 20?– What is 21?– What is 22?– What is 23?

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Reading a Binary Number

• In general, a binary number is represented like this:

bn-1 bn-2 … b2 b1 b0

• This binary number can be translated to decimal using this formula:

bn-1*2n-1 + bn-2*2n-2 + … + b2 * 22 + b1 * 21 + b0 * 20.

• As a quick review: – What is 20? 1– What is 21? 2– What is 22? 4– What is 23? 8

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Translating from Binary to Decimal

• In general, a binary number is represented like this:

bn-1 bn-2 … b2 b1 b0

• This binary number can be translated to decimal using this formula:

bn-1*2n-1 + bn-2*2n-2 + … + b2 * 22 + b1 * 21 + b0 * 20.

• So, how do we translate binary number 10011 to decimal?

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Translating from Binary to Decimal

• In general, a binary number is represented like this:

bn-1 bn-2 … b2 b1 b0

• This binary number can be translated to decimal using this formula:

bn-1*2n-1 + bn-2*2n-2 + … + b2 * 22 + b1 * 21 + b0 * 20.

• So, how do we translate binary number 10011 to decimal?

1*24 + 0*23 + 0*22 + 1*21 + 1*20 = 16 + 2 + 1 = 19.

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Translating from Binary to Decimal

• In general, a binary number is represented like this:

bn-1 bn-2 … b2 b1 b0

• This binary number can be translated to decimal using this formula:

bn-1*2n-1 + bn-2*2n-2 + … + b2 * 22 + b1 * 21 + b0 * 20.

• How do we translate binary number 101000 to decimal?

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Translating from Binary to Decimal

• In general, a binary number is represented like this:

bn-1 bn-2 … b2 b1 b0

• This binary number can be translated to decimal using this formula:

bn-1*2n-1 + bn-2*2n-2 + … + b2 * 22 + b1 * 21 + b0 * 20.

• How do we translate binary number 101000 to decimal?

1*25 + 0*24 + 1*23 + 0*22 + 0*21 + 0*20 = 32 + 8 = 40.

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binaryToDecimal Function

• Let's write a function binaryToDecimal that translates a binary number into a decimal number.

• What arguments should the function take?• What type should the function return?

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binaryToDecimal Function

• Let's write a function binaryToDecimal that translates a binary number into a decimal number.

• What arguments should the function take?– The binary number can be represented as a string.– It can also be represented as an int.– We will go with the string choice.

• What type should the function return? – The decimal number should be an int.

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binaryToDecimal Function public static int binaryToDecimal(String text) { int result = 0; for (int i = 0; i < text.length(); i++) { String c = text.substring(i, i+1); if ("01".indexOf(c) == -1) { System.out.printf("Error: invalid binary number %s,

exiting...\n", text); System.exit(0); } int digit = Integer.parseInt(c); int power_of_2 = (int) (Math.pow(2, text.length() - i - 1)); result = result + digit * power_of_2; } return result; }

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Translating from Decimal to Binary

• How can we translate a decimal number, such as 37, to binary?

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Translating from Decimal to Binary

• How can we translate a decimal number, such as 37, to binary?

• Answer: we repeatedly divide by 2, until the result is 0, and use the remainder as a digit.

• 37 / 2 = 18, with remainder ???

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Translating from Decimal to Binary

• How can we translate a decimal number, such as 37, to binary?

• Answer: we repeatedly divide by 2, until the result is 0, and use the remainder as a digit.

• 37 / 2 = 18, with remainder 1. So, b0 is 1.• 18 / 2 = 9, with remainder ???

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Translating from Decimal to Binary

• How can we translate a decimal number, such as 37, to binary?

• Answer: we repeatedly divide by 2, until the result is 0, and use the remainder as a digit.

• 37 / 2 = 18, with remainder 1. So, b0 is 1.

• 18 / 2 = 9, with remainder 0. So, b1 is 0.• 9 / 2 = 4, with remainder ???

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Translating from Decimal to Binary

• How can we translate a decimal number, such as 37, to binary?

• Answer: we repeatedly divide by 2, until the result is 0, and use the remainder as a digit.

• 37 / 2 = 18, with remainder 1. So, b0 is 1.

• 18 / 2 = 9, with remainder 0. So, b1 is 0.

• 9 / 2 = 4, with remainder 1. So, b2 is 1.• 4 / 2 = 2, with remainder ???

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Translating from Decimal to Binary

• How can we translate a decimal number, such as 37, to binary?

• Answer: we repeatedly divide by 2, until the result is 0, and use the remainder as a digit.

• 37 / 2 = 18, with remainder 1. So, b0 is 1.

• 18 / 2 = 9, with remainder 0. So, b1 is 0.

• 9 / 2 = 4, with remainder 1. So, b2 is 1.

• 4 / 2 = 2, with remainder 0. So, b3 is 0.• 2 / 2 = 1, with remainder ???

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Translating from Decimal to Binary

• How can we translate a decimal number, such as 37, to binary?

• Answer: we repeatedly divide by 2, until the result is 0, and use the remainder as a digit.

• 37 / 2 = 18, with remainder 1. So, b0 is 1.

• 18 / 2 = 9, with remainder 0. So, b1 is 0.

• 9 / 2 = 4, with remainder 1. So, b2 is 1.

• 4 / 2 = 2, with remainder 0. So, b3 is 0.

• 2 / 2 = 1, with remainder 0. So, b4 is 0.• 1 / 2 = 0, with remainder ???

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Translating from Decimal to Binary

• How can we translate a decimal number, such as 37, to binary?

• Answer: we repeatedly divide by 2, until the result is 0, and use the remainder as a digit.

• 37 / 2 = 18, with remainder 1. So, b0 is 1.

• 18 / 2 = 9, with remainder 0. So, b1 is 0.

• 9 / 2 = 4, with remainder 1. So, b2 is 1.

• 4 / 2 = 2, with remainder 0. So, b3 is 0.

• 2 / 2 = 1, with remainder 0. So, b4 is 0.

• 1 / 2 = 0, with remainder 1. So, b5 is 1.• So, 37 in binary is ???

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Translating from Decimal to Binary

• How can we translate a decimal number, such as 37, to binary?

• Answer: we repeatedly divide by 2, until the result is 0, and use the remainder as a digit.

• 37 / 2 = 18, with remainder 1. So, b0 is 1.

• 18 / 2 = 9, with remainder 0. So, b1 is 0.

• 9 / 2 = 4, with remainder 1. So, b2 is 1.

• 4 / 2 = 2, with remainder 0. So, b3 is 0.

• 2 / 2 = 1, with remainder 0. So, b4 is 0.

• 1 / 2 = 0, with remainder 1. So, b5 is 1.• So, 37 in binary is 100101.

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decimalToBinary Function

• Let's write a function decimalToBinary that translates a decimal number into a binary number.

• What arguments should the function take?

• What type should the function return?

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decimalToBinary Function

• Let's write a function decimalToBinary that translates a decimal number into a binary number.

• What arguments should the function take?– The decimal number should be an int.

• What type should the function return? – Again, the most simple choice is to represent the binary

number as a string.

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decimalToBinary Function public static String decimalToBinary(int number) { String result = ""; while(true) { int remainder = number % 2; String digit = Integer.toString(remainder); result = digit + result; number = number / 2; if (number == 0) { break; } } return result; }

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Hexadecimal Numbers• Another very popular way to represent numbers

among programmers is hexadecimal numbers.• Hexadecimal numbers use 16 digits:

0123456789abcdef

• a stands for 10.• b stands for 11.• c stands for 12.• d stands for 13.• e stands for 14.• f stands for 15.

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Reading a Hexadecimal Number

• A hexadecimal number is represented like this:

hn-1 hn-2 … h2 h1 h0

• Each digit hi is one of the sixteen digits:– 0123456789abcdef.

• For example, consider hexadecimal number 5b7.• What is n?• What is h0?

• What is h1?

• What is h2?

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Reading a Hexadecimal Number

• A hexadecimal number is represented like this:

hn-1 hn-2 … h2 h1 h0

• Each digit hi is one of the sixteen digits:– 0123456789abcdef.

• For example, consider hexadecimal number 5b7.• What is n? 3• What is h0? 7

• What is h1? b

• What is h2? 5

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Reading a Hexadecimal Number

• A hexadecimal number is represented like this:

hn-1 hn-2 … h2 h1 h0

• This hexadecimal number can be translated to decimal using this formula:

bn-1*16n-1 + bn-2*16n-2 + … + b2 * 162 + b1 * 161 + b0*160.

• Note: if bi is one of abcdef, use the corresponding number (10, 11, 12, 13, 14, 15) when you plug it into the formula.

• This looks like the formula for binary numbers, except that here we use powers of 16 instead of powers of 2.

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From Hexadecimal to Decimal

• A hexadecimal number is represented like this:

hn-1 hn-2 … h2 h1 h0

• This hexadecimal number can be translated to decimal using this formula:

bn-1*16n-1 + bn-2*16n-2 + … + b2 * 162 + b1 * 161 + b0*160.

• So, how do we translate hexadecimal number 5b7 to decimal?

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From Hexadecimal to Decimal

• A hexadecimal number is represented like this:

hn-1 hn-2 … h2 h1 h0

• This hexadecimal number can be translated to decimal using this formula:

bn-1*16n-1 + bn-2*16n-2 + … + b2 * 162 + b1 * 161 + b0*160.

• So, how do we translate hexadecimal number 5b7 to decimal? (note that we plug in 11 for b).

5*162 + 11*161 + 7*160 = 5*256 + 11*16 + 7 = 1463.

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From Hexadecimal to Decimal

• A hexadecimal number is represented like this:

hn-1 hn-2 … h2 h1 h0

• This hexadecimal number can be translated to decimal using this formula:

bn-1*16n-1 + bn-2*16n-2 + … + b2 * 162 + b1 * 161 + b0*160.

• So, how do we translate hexadecimal number bad to decimal?

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From Hexadecimal to Decimal

• A hexadecimal number is represented like this:

hn-1 hn-2 … h2 h1 h0

• This hexadecimal number can be translated to decimal using this formula:

bn-1*16n-1 + bn-2*16n-2 + … + b2 * 162 + b1 * 161 + b0*160.

• So, how do we translate hexadecimal number bad to decimal?

11*162 + 10*161 + 13*160 = 11*256 + 10*16 + 13 = 2989.

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hexToDecimal Function

• Let's write a function hexToDecimal that translates a hexadecimal number into a decimal number.

• What arguments should the function take?• What type should the function return?

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hexToDecimal Function

• Let's write a function binaryToDecimal that translates a binary number into a decimal number.

• What arguments should the function take?– The hexadecimal number can be represented as a string.

• What type should the function return? – The decimal number should be an int.

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hexToDecimal Function public static String decimalToHex(int number) { String result = ""; String digits = "0123456789abcdef";

while(true) { int remainder = number % 16; String digit = digits.substring(remainder, remainder+1); result = digit + result; number = number / 16; if (number == 0) { break; } } return result; }

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From Decimal to Hexadecimal

• How can we translate a decimal number, such as 540, to hexadecimal?

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From Decimal to Hexadecimal

• How can we translate a decimal number, such as 540, to hexadecimal?

• Answer: we repeatedly divide by 16, until the result is 0, and use the remainder as a digit.

• If the remainder is 10, 11, 12, 13, 14, or 15, we use respectively digit a, b, c, d, e, or f.

• 540 / 16 = 33, with remainder 12. So, h0 is ???

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From Decimal to Hexadecimal

• How can we translate a decimal number, such as 540, to hexadecimal?

• Answer: we repeatedly divide by 16, until the result is 0, and use the remainder as a digit.

• If the remainder is 10, 11, 12, 13, 14, or 15, we use respectively digit a, b, c, d, e, or f.

• 540 / 16 = 33, with remainder 12. So, h0 is c.• 33 / 16 = ??? with remainder ???

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From Decimal to Hexadecimal

• How can we translate a decimal number, such as 540, to hexadecimal?

• Answer: we repeatedly divide by 16, until the result is 0, and use the remainder as a digit.

• If the remainder is 10, 11, 12, 13, 14, or 15, we use respectively digit a, b, c, d, e, or f.

• 540 / 16 = 33, with remainder 12. So, h0 is c.

• 33 / 16 = 2 with remainder 1. So, h1 is 1.• 2 / 16 = ??? with remainder ???

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From Decimal to Hexadecimal

• How can we translate a decimal number, such as 540, to hexadecimal?

• Answer: we repeatedly divide by 16, until the result is 0, and use the remainder as a digit.

• If the remainder is 10, 11, 12, 13, 14, or 15, we use respectively digit a, b, c, d, e, or f.

• 540 / 16 = 33, with remainder 12. So, h0 is c.

• 33 / 16 = 2 with remainder 1. So, h1 is 1.

• 2 / 16 = 0 with remainder 2. So, h2 is 2.• So, 540 in hexadecimal is ???

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From Decimal to Hexadecimal

• How can we translate a decimal number, such as 540, to hexadecimal?

• Answer: we repeatedly divide by 16, until the result is 0, and use the remainder as a digit.

• If the remainder is 10, 11, 12, 13, 14, or 15, we use respectively digit a, b, c, d, e, or f.

• 540 / 16 = 33, with remainder 12. So, h0 is c.

• 33 / 16 = 2 with remainder 1. So, h1 is 1.

• 2 / 16 = 0 with remainder 2. So, h2 is 2.• So, 540 in hexadecimal is 21c.

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decimalToHexadecimal Function

• Let's write a function decimalToHexadecimal that translates a decimal number into a binary number.

• What arguments should the function take?

• What type should the function return?

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decimalToHexadecimal Function

• Let's write a function decimalToHexadecimal that translates a decimal number into a binary number.

• What arguments should the function take?– The decimal number should be an int.

• What type should the function return? – The hexadecimal number should be a string.

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decimalToHexadecimal Function public static String decimalToHexadecimal(int number) { String result = ""; String digits = "0123456789abcdef";

while(true) { int remainder = number % 16; String digit = digits.substring(remainder, remainder+1); result = digit + result; number = number / 16; if (number == 0) { break; } } return result; }